Question: Simplify and expand the following expression: $ \dfrac{3}{k - 2}- \dfrac{3}{k + 7}- \dfrac{1}{k^2 + 5k - 14} $
Answer: First find a common denominator by finding the least common multiple of the denominators. Try factoring the denominators. We can factor the quadratic in the third term: $ \dfrac{1}{k^2 + 5k - 14} = \dfrac{1}{(k - 2)(k + 7)}$ Now we have: $ \dfrac{3}{k - 2}- \dfrac{3}{k + 7}- \dfrac{1}{(k - 2)(k + 7)} $ The least common multiple of the denominators is: $ (k - 2)(k + 7)$ In order to get the first term over $(k - 2)(k + 7)$ , multiply by $\dfrac{k + 7}{k + 7}$ $ \dfrac{3}{k - 2} \times \dfrac{k + 7}{k + 7} = \dfrac{3(k + 7)}{(k - 2)(k + 7)} $ In order to get the second term over $(k - 2)(k + 7)$ , multiply by $\dfrac{k - 2}{k - 2}$ $ \dfrac{3}{k + 7} \times \dfrac{k - 2}{k - 2} = \dfrac{3(k - 2)}{(k - 2)(k + 7)} $ Now we have: $ \dfrac{3(k + 7)}{(k - 2)(k + 7)} - \dfrac{3(k - 2)}{(k - 2)(k + 7)} - \dfrac{1}{(k - 2)(k + 7)} $ $ = \dfrac{ 3(k + 7) - 3(k - 2) - 1} {(k - 2)(k + 7)} $ Expand: $ = \dfrac{3k + 21 - 3k + 6 - 1}{k^2 + 5k - 14} $ $ = \dfrac{26}{k^2 + 5k - 14}$